U.S. Pat. No. 4,901,839 granted Feb. 20, 1990 to G. E. Johnson, et al. discloses a self adjustable escalator handrail drive which utilizes drive rollers which are eccentrically mounted in rotatable bearings. The drive may include two pairs of drive rollers which are driven by a chain and sprocket assembly. The sprockets are mounted on each drive roller shaft, and the drive chain is entrained around each of the sprockets and around a drive sprocket connected to an electric motor. The drive roller pairs serially engage the handrail in a mangle-like arrangement. This arrangement can successfully drive a considerable load, but does display certain inefficiencies due to the use of the single chain to drive four roller sprockets. The drive roller set closest to the drive sprocket runs at a higher efficiency then the drive roller set further from the drive sprocket. It would be desirable to be able to increase the drive power of this type of self-tightening handrail drive to be able to drive handrails on longer moving walkways, and to drive handrails on curved escalators, which involve higher friction loads due to the curvature in plan of the path of travel of the handrail.